How can I solidify a 2d shape so that the rim isn't solidified and the center is solidified the most ? (Think creating a dome from a sphere)
A way to do that using a modifier is to shrinkwrap the jewel to a plane, using a weighted vertex group (the plane here is displayed in wireframe mode):



Select the vertices you want to move (all except the borders).

Go to weight paint. 

Validate 'vertex selection masking'

Enable X mirror

Then paint (add) the wanted vertices.

Once done, smooth the result.



After that, you can tune the influence in the modifier :





Following the comment, in order to use more easily a mirror, start from a circle, extrude inside then grid fill the inner part. That will allow you to have an edge loop or more around the jewel, so that you can connect easily the main part to the mirror center. Here is the kind of start that can be good for that :